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Theorem issiga 29952
Description: An alternative definition of the sigma-algebra, for a given base set. (Contributed by Thierry Arnoux, 19-Sep-2016.)
Assertion
Ref Expression
issiga (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
Distinct variable groups:   𝑥,𝑂   𝑥,𝑆

Proof of Theorem issiga
Dummy variables 𝑜 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6178 . . . 4 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑂 ∈ V)
2 elex 3198 . . . 4 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ V)
31, 2jca 554 . . 3 (𝑆 ∈ (sigAlgebra‘𝑂) → (𝑂 ∈ V ∧ 𝑆 ∈ V))
43a1i 11 . 2 (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) → (𝑂 ∈ V ∧ 𝑆 ∈ V)))
5 simpr1 1065 . . . . 5 ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → 𝑂𝑆)
6 elex 3198 . . . . 5 (𝑂𝑆𝑂 ∈ V)
75, 6syl 17 . . . 4 ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → 𝑂 ∈ V)
87a1i 11 . . 3 (𝑆 ∈ V → ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → 𝑂 ∈ V))
98anc2ri 580 . 2 (𝑆 ∈ V → ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → (𝑂 ∈ V ∧ 𝑆 ∈ V)))
10 df-siga 29949 . . . 4 sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))})
11 sigaex 29950 . . . 4 {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))} ∈ V
12 pweq 4133 . . . . . . 7 (𝑜 = 𝑂 → 𝒫 𝑜 = 𝒫 𝑂)
1312sseq2d 3612 . . . . . 6 (𝑜 = 𝑂 → (𝑠 ⊆ 𝒫 𝑜𝑠 ⊆ 𝒫 𝑂))
14 sseq1 3605 . . . . . 6 (𝑠 = 𝑆 → (𝑠 ⊆ 𝒫 𝑂𝑆 ⊆ 𝒫 𝑂))
1513, 14sylan9bb 735 . . . . 5 ((𝑜 = 𝑂𝑠 = 𝑆) → (𝑠 ⊆ 𝒫 𝑜𝑆 ⊆ 𝒫 𝑂))
16 eleq12 2688 . . . . . 6 ((𝑜 = 𝑂𝑠 = 𝑆) → (𝑜𝑠𝑂𝑆))
17 simpr 477 . . . . . . 7 ((𝑜 = 𝑂𝑠 = 𝑆) → 𝑠 = 𝑆)
18 difeq1 3699 . . . . . . . . . 10 (𝑜 = 𝑂 → (𝑜𝑥) = (𝑂𝑥))
1918adantr 481 . . . . . . . . 9 ((𝑜 = 𝑂𝑠 = 𝑆) → (𝑜𝑥) = (𝑂𝑥))
2019eleq1d 2683 . . . . . . . 8 ((𝑜 = 𝑂𝑠 = 𝑆) → ((𝑜𝑥) ∈ 𝑠 ↔ (𝑂𝑥) ∈ 𝑠))
21 eleq2 2687 . . . . . . . . 9 (𝑠 = 𝑆 → ((𝑂𝑥) ∈ 𝑠 ↔ (𝑂𝑥) ∈ 𝑆))
2221adantl 482 . . . . . . . 8 ((𝑜 = 𝑂𝑠 = 𝑆) → ((𝑂𝑥) ∈ 𝑠 ↔ (𝑂𝑥) ∈ 𝑆))
2320, 22bitrd 268 . . . . . . 7 ((𝑜 = 𝑂𝑠 = 𝑆) → ((𝑜𝑥) ∈ 𝑠 ↔ (𝑂𝑥) ∈ 𝑆))
2417, 23raleqbidv 3141 . . . . . 6 ((𝑜 = 𝑂𝑠 = 𝑆) → (∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ↔ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆))
25 pweq 4133 . . . . . . . 8 (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
26 eleq2 2687 . . . . . . . . 9 (𝑠 = 𝑆 → ( 𝑥𝑠 𝑥𝑆))
2726imbi2d 330 . . . . . . . 8 (𝑠 = 𝑆 → ((𝑥 ≼ ω → 𝑥𝑠) ↔ (𝑥 ≼ ω → 𝑥𝑆)))
2825, 27raleqbidv 3141 . . . . . . 7 (𝑠 = 𝑆 → (∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠) ↔ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
2928adantl 482 . . . . . 6 ((𝑜 = 𝑂𝑠 = 𝑆) → (∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠) ↔ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
3016, 24, 293anbi123d 1396 . . . . 5 ((𝑜 = 𝑂𝑠 = 𝑆) → ((𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)) ↔ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
3115, 30anbi12d 746 . . . 4 ((𝑜 = 𝑂𝑠 = 𝑆) → ((𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠))) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
3210, 11, 31abfmpel 29294 . . 3 ((𝑂 ∈ V ∧ 𝑆 ∈ V) → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
3332a1i 11 . 2 (𝑆 ∈ V → ((𝑂 ∈ V ∧ 𝑆 ∈ V) → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))))
344, 9, 33pm5.21ndd 369 1 (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cdif 3552  wss 3555  𝒫 cpw 4130   cuni 4402   class class class wbr 4613  cfv 5847  ωcom 7012  cdom 7897  sigAlgebracsiga 29948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-siga 29949
This theorem is referenced by:  baselsiga  29956  sigasspw  29957  issgon  29964  isrnsigau  29968  dmvlsiga  29970  pwsiga  29971  prsiga  29972  sigainb  29977  insiga  29978  sigapildsys  30003  imambfm  30102  carsgsiga  30162
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